On the application of mixed finite element methods to the wave equations

M2AN. M

- M

T ^UNC G ^EVECI

On the application of mixed finite element methods to the wave equations

M2AN. Mathematical modelling and numerical analysis - Modéli- sation mathématique et analyse numérique, tome 22, n^o2 (1988), p. 243-250

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MOOÉUSATIOH MATHÉMATIQUE ET ANALYSE NUMÉRIQUE (Vol. 22, ne 2, 1988, p. 243 à 250)

ON THE APPLICATION OF MIXED FINITE ELEMENT METHODS TO THE WAVE EQUATIONS (*)

by Tune GEVECI O

Abstract — The convergence of certain semidiscrete approximation schemes based on the

« velocity-stress » formulation of the wave équation and spaces such as those tntroduced by Raviart and Thomas is discussed The discussion also apphes to similar schemes for the équations of elasticity

Resumé. — La convergence de certains schémas d'approximation semi-discrète basés sur la formulation « vitesse-contrainte » de l'équation d'onde et d'espace tel que ceux introduits par Raviart et Thomas est discuté La discussion s'applique également pour les schémas similaires aux équations d'élasticité

1. THE « VELOCITY-STRESS » FORMULATION OF THE WAVE EQUATION AND A SEMIDISCRETE VERSION

Let us consider the following initial-boundary value problem for the wave équation :

D f u ( t , x ) ~ & u ( t , x ) = f ( t9x ) , r > 0 , x e O c R 2 , ( 1 . 1 ) « ( * , * ) = 0 , t > 0 , xeT,

u ( 09x ) = uo( x ) , Dtu ( 09x ) = vo( x ) , x e H ,

where fl is a bounded domain with boundary F, and ƒ, u⁰, v⁰ are given functions. Introducing the « stress » cr = Vu, (1.1) may be reformulated as

Dfu(t,x) - d i v a ( f , j t ) = f(t,x) , t>0 , J C Ë O ,

(1.2) u(t9x) = 0, t>0, xeT,

M ( 0 , X) = uo(x) , £>, w(0, x) = vo(x) ,

(*) Received in October 1986.

(!) Department of Mathematical Sciences, San Diego State University San Diego, Califorma 92182

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244 T. GEVECI

We use the notation of Johnson and Thomée [12] :

V = L2(n) , H = {x e L2(nf : div x e L2(H)} . Using Green's formula

f [ Ç

u div x dx = u\ . n ds — Vu . x dx Jn Jr Jn

where n is the unit exterior normal to F, a Galerkin version of (1.2) is to seek u(t) e V, <r(t) e H, t > 0, satisfying

(Dfu(t),w)- ( d i v a ( 0 , w ) = ( / ( r ) , w ) , weV , (1.3) ( ^ ( O x ) + ( M ( O>d i vx) = 0 , xe / f ,

u(0) = MO, Dfw(0) = t?0>

where the parentheses dénote the appropriate inner products (L2-inner product in V, L2(n)2-inner product in H). \ïVh<^V and Hha H are finite dimensional subspaces, such as the spaces introduced by Raviart and Thomas [13], and by Brezzi, Douglas, Jr. and Marini [6], a semidiscrete version of (1.3) seeks uh{t) G Vh, <rh(t) e Hh, t > 0, satisfying

(Dfuh(t), wh) - (divaA(0, wh) = (f(t), wh) , wheVk,

uh(0) = M0 ) h, Dtuh(0) = i ;0 i h s

whcrc uOjk, voh e Vh are approximations to u0 and v0, respectively.

Johnson and Thomée [12] have discussed the parabolic counterpart of (1.3A). The analysis of convergence of (1.3/,) can be carried out along similar Unes, parallel to Baker and Bramble [4], for example. (1.3A) is treated essentially as a non-conforming « displacement » model for the wave équation (1.1). The purpose of this note is to discuss the convergence of the

« velocity-stress » models based on pairs of spaces (Vh, Hh) such as those in [6], [12], [13]. Thus, defining v = Dtu, vh = Dtuh, (1.3) and (1.3A) are transformed, respectively, to

( Dtv ( t ) , w ) ~ ( d i v a ( 0 , w ) = ( f ( t ) , w ) , w e V ,

(1.4) ( D , a ( O , x ) + ( » ( O > d i vX) = 0 , X e H , v(0) = v0, CT(0) = a0 = Vw0 , where v(t) eV, <r(t)e H, t^ 0, and

(Dt vh(t), wh) - (div *h(t), wh) = (ƒ(0, wh)9 wheVh,

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(1.4,) (A^(O,Xfc)+(»A(O,divX f c) = 0 , Xh^Hh9

where vh(t) e Vh, vh(t) e Hh, t a* 0.

We now list the basic features of the space V^h, H^h which lead to a straightforward analysis of the convergence of vh to v and crh to a :

(H.l) There exists a linear operator Uh: H ^ Hh such that (1.5) ( d i v I I^{h X}, w * ) = (div^X,w^h) Vw^heV^h, x^H, (1.6) \ \ n ^{h X}~ x \ \ ^ C h^s\ \ x \ \^s f o r l ^ s ^ r , r ^ 2 (|| || is the L²(n)²-norm, and ||.||⁵ is the 7/*(n)²-norm).

(H.2) There exists a linear operator Ph : V -> Vh such that (1.7) ( P^hv , d i v^{X h}) = ( v⁹d i v^{X h}) V ^{X A} e ^ ⁵ veV, (1.8) I I ^ A » - Ü | | «CA'llüH,, U ^ r , r ^ 2

(|| || is the L2(n)-norm, and || ||s is the Hs(ü,)-norm, and, as usual, C dénotes a generic constant which dépends only on the data and on the particular discretization scheme).

If ft is a polygonal domain and V^h, H^h are the Raviart-Thomas spaces [12], [13], or if these spaces are the pairs introduced in the paper by Brezzi, Douglas, Jr., and Marini [6], div \h e Vh-> an<^ Ph c a n ^e t a^en to be the L²-projection. For an example of a pair (V^h, H^h) satisfying the above hypotheses (with r = 2), where Ph is not the L2-projection, we refer the reader to the paper by Johnson and Thomée [12]. We would also like to point out that (H.l) and (H.2) are valid for the mixed method that has been introduced by Arnold, Douglas, Jr., and Gupta [3] to approximate solution of plane elasticity problems. Our analysis is readily adapted to the corresponding (genuine) velocity-stress formulation of the time-dependent problem.

We can now state and prove our convergence resuit :

THEOREM : If u is the solution of (1.1), v = Dtu, <J = Vw, and if the pair {vh> ah} w tne solution of(lAh), under the hypotheses (H.l) and (H.2) we have, for 1 =s s === r, r ^ 2,

(1.9) | | » * ( 0 -

vol. 22, n° 2, 1988

246 T GEVECI

Proof : Let us dénote by X the space V x H, the éléments of which will be designated as £ = {v, o-} or Ç = {w, x} and set

( ( € , 5 ) ) = (v w)+(<T,^x),

min

Let Xh = Vh x Hh be equipped with ((. , . ) ) and the induced norm III • III . Eléments of Xh will be designated as £h = {vh,<rh} or £A = {wh, \h} •

We define the bilinear form a{. , . ) on X by (1.10) o(Ê, £) = - (diva, w ) + (t),div^X) for £ = {i>, a } , 4 = {w, x } •

We can now express (1.4) as

(1.11) ((D, * ( * ) , £ ) ) + « ( € ( 0 . 0 = ( / ( 0 , w ) , U ^ r ( 4 ( 0 = { r ( f ) , a ( O } , C = { w , x } ) , and we can express {i.A^h) as

A) = (/(O, wfc) , CheXh

Let us define P^hk= {Ph ^v> ⁿA ^a} f °^r € = {^v> "•} e ^T, and observe that (1-12) fl(£*c,t*)

by ( H . l ) ((1.5)) and (H.2) ((1.7)).

Therefore we obtain from (1.11) (1.13) ((£>, Ph

= (/(O, w») + ((P

A 6(0 - D, Ç(0, 5*)) ,

Setting e^A(f ) = P^h 6(r) - 6*(0, (1-Hft) and (1.13) yield

(1.14) ((A s*(0, t*)) + a(e*(0. 5») =

Let us define Ah : Xh -> A!"fc by (1.15)

Since (1.16)

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as is readily seen (cf. (1.10)), Ah is shew-adjoint, (1.17)

and -Ah générâtes the unitary group e \ In particular, (1.18) III e~'Ahih(0) III = BI Êh(0)111, teU.

Let us dénote by P°h:X^>Xh the projection with respect ((. , . ) ) . We can now express (1.14) as

(1.19) D, 8,(0 + A„ e*(0 = P°h(Ph Dt UO - D, so that

(1.20) e,(0 = e-/Afte,(0)+ f e~ « " ^ P l ( Ph Dj € ( T) _ Dj €(T)) dr . Jo

(1.18) and (1.20) yield the estimate

( 1 . 2 1 ) III 8 , ( 0 III ^ III 6 , ( 0 ) III + f III Ph DT 6 ( T ) - Z)T 6 ( T ) III ^ T Jo

(El is the ((. , .))-projection).

(1.21) is readily translated to

« C ( | | ü0- f o , f t | | + 11 °"o — °-0,ftli + H-PA^O - üoll + I I H A ^ O - O ' O I I

+ | ' (||Pf cDTo(T)-DT»(T)|| + ||II,,£>Ta(T)-Z>T<T<

Jo

and this, together with (1.6) and (1.8), yields (1.22) | | P * i > ( 0 - M 0 | | + | | n , a ( 0 - c 7 , ( 0 | |

)-V0,h\\ + II a0 - ar0> h || + ^ ( | K HS

vol. 22, n° 2, 1988

248 T. GEVECI

Since

||»(O-»*(OII + 1^(0-^(011

=s \\v(t)-Phv(t)\\ + \\Phv(t)-vh(t)\\

+ | | a ( 0 - nh CT(0|| + \\nha(t)-<rh(t)\\

by (1.6) and (1.8), and

n»(on,*Kii

+ r

J

*IKII,+ l

J o

p

Jo

(1.22) leads to (1.9), the assertion of the theorem.

2. SOME OBSERVATIONS IN REGARD TO THE TEME-DIFFERENCING OF THE SEMIDISCRETE MODEL

(1.4/,) leads to a system of ordinary differential équations in the form M^ö D^t W - DX = F ,

where W corresponds to v^h, 2 corresponds to v^h, M^o, M¹ are symmetrie, positive-definite matrices, and DT dénotes the transpose of D. The application of impiicit Euier time-differencing

Mo

'° k

(2.2) yn + 1 _ vn

(k dénotes the time step), nécessitâtes the solution of

n + 1 = kFn + l + M0Wn ,

Mo is in block-diagonal form if Vh consists of functions with no continuity requirement across inter-element boundaries, as is the case in [6], [12], [13], and the élimination of Wn +1 in (2.3) is efficiently implementable. This leads to a System in the form

(2.4)

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where M¹ is symmetrie, positive definite and D^TMQ¹D is symmetrie, positive-semidefinite, for the détermination of 2" ^{+ 1}.

On the other hand, (1.3*) leads to

M0DfU-DX = F,

K 0) MX

where U corresponds to uh. (2.5) can be expressed as (2.6) M^o Df U + DMï ¹D^TU^F ,

where M^o, DMf¹ D^T are symmetrie, positive-definite [12]. If (2.6) is expressed as a System in {U, W} ,

D^t U - W = 0

Mö Dt W + Z)Mf lDTU=F , and implicit Euler time-differencing is applied to (2.7),

k

(2.8) w n + 1 Tx^n

élimination of Wrt + 1 leads to a System in the form (2.9) (Mo + k2DMï 1DT)Un + 1 = G

The matrix in (2.9) is symmetrie, positive-definite, so that (2.9) is solvable.

But Afj is not block-diagonal, unlike Mo, so that deriving the reduced system (2.9), which includes inverting Ml5 is more expensive than forming the reduced system (2.4). The time-independent counterpart of (2.5),

- D2 = F ,

( 2*^{1 0 )} M¹S + D^rL7 = 0 ,

led Arnold and Brezzi [2] to relax the requirement that div o^ e L2(lï) in order to have a block-diagonal matrix instead of Mx and be able to eliminate 2 efficiently. This approach has to introducé a multiplier corresponding to the relaxation of the requirement div ah e L2(H).

The above considérations suggest that the « velocity-stress » formulation (1.4,,) may be préférable to (1.3,,) if the approximation of the « stress » a is of primary concern.

The application of diagonally implicit Runge-Kutta methods (see, for example, Crouzeix [8], Crouzeix and Raviart [9], Alexander [1], Burrage

vol. 22, ne 2, 1988

250 T. GEVECI

[7], Dougalis and Serbin [10]) to (2.1) leads to Systems similar to (2.4) so that our discussion is relevant to higher-order time differencing as well. We will not prove error estimâtes for such full-descrete approximation schemes based on (lAk). Such estimâtes should be obtainable by employing techniques that have been utilized in [5] or [11], for example.

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[1] R. ALEX ANDER, Diagonally implicity Runge-Kutta methods for stijfO.D.E.'s, SIAM J. Numer. Anal. 14 (1977), 1006-1021.

[2] D . N. ARNOLD and F. BREZZI, Mixed and nonconforming finite element methods : Implementation, postprocessing and error estimâtes, R.A.LR.O.

Math. Model, and Num. Anal. (M2AN) 1 (1985), 7-32.

[3] D. N. ARNOLD, J. D O U G L A S , Jr., and C. P. GUPTA, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), 1-22.

[4] G. A. BAKER and J. H. BRAMBLE, Semidiscrete and single step fully discrete approximations for second order hyperbolic équations, R.A.LR.O. Anal. Num.

13 (1979), 75-100.

[5] P. BRENNER, M. CROUZEIX and V. THOMÉE, Single step methods for inhomogeneous linear differential équations in Banach spaces, R.A.LR.O.

Anal. Num. 16 (1982), 5-26.

[6] F. BREZZI, J. D O U G L A S , Jr. and L. D. MARINI, TWO families of mixed finite éléments for second order elliptic problems, Numer. Math. 47 (1985), 217-235.

[7] K. BURRAGE, Efficiently implementatie algebraically stable Runge-Kutta methods, SIAM J. Numer. Anal. 19 (1982), 245-258.

[8] M. CROUZEIX, Sur l'approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta, thèse, Paris (1975).

[9] M. CROUZEIX and P.-A. RAVIART, Approximation des problèmes d'évolution, preprint, Université de Rennes (1980).

[10] V. DOUGALIS and S. M. SERBIN, On some unconditionally stable, higher order methods for numerical solution o f the structural dynamics équations, Int. J.

Num. Meth. Eng. 18 (1982), 1613-1621.

[11] E. GEKELER, Discretization Methods for Stable Initial Value Problems, Springer Lecture Notes in Mathematics 1044 (1984), Springer-Verlag, Berlin, Heidel- berg, New York,

[12] C. JOHNSON and V. THOMÉE, Error estimâtes for some mixed finite element methods for parabolic type problems, R.A.LR.O. Anal. Num. 15 (1981), 41-78.

[13] P.-A. RAVIART and J. M. THOMAS, A mixed finite element method for 2nd order problems, in Mathematical Aspects o f the Finite Element Method, Springer Lecture Notes in Mathematics 606 (1977), Springer-Verlag, Berlin-Heidelberg- New York.

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